How do I make an animation from a curve?
Select your object and Export as .MDD - you might have to enable it under preferences. Apply the Curve Modifier to your object.

In a new file append your object and remove any animation curves from the Graph Editor. With the object selected Import the .MDD file - you will see per frame that curves have been added to the graph editor. You might have to move the frames to correspond to the peaks of the action. In my example I just started before the first frame and stopped before the last...



Update: While the above method does work, there is also Alembic .abc export which you might try. Export the number of frames you want - when you import the .abc file in a new Blender you will see the animation is baked in. There are no key frames in this export.